A Frobenius homomorphism for Lusztig's quantum groups for arbitrary roots of unity (Q2800062)

Fix a finite-dimensional semisimple Lie algebra \(g\) and a primitive \(l\)-th root of unity \(q\). For these data Lusztig defined a complex Hopf algebra \(U^{L}_q(g)\). For \(l\) odd (and not divisible by \(3\) in the case \(g=G_2\)) Lusztig defined a Hopf algebra homomorphism \(\mathrm{Frob}: U^{L}_q(g)\rightarrow U(g)\). This homomorphism is related to Frobenius homomorphism in the case when \(l\) is prime. The kernel \(u_q^L(g)\) of this homomorphism is called the small quantum group of Frobenius-Lusztig kernel. It is a finite dimensional Hopf algebra.NEWLINENEWLINEIn the paper under review the case of an arbitrary root of unity is considered. An analog of the Frobenius homomorphism with finite-dimensional kernel is defined. The image of this homomorphism is not \(U(g)\) but \(U(g^{(l)})\), where \(g^{(l)}\) is a Lie algebra with a dual root system in the cases \(2,3\mid l\). This analog has a finite dimensional kernel which is explicitly described.